Effect of size and position of supporting buoys on the dynamics of spread mooring systems

ABSTRACT

Vessels moored in deep water may require buoys to

support

part of the weight of the mooring lines.

The effects that

size, and location of supporting buoys have on the dynamics

of Spread Mooring Sysems SMS at different water depths

are assessed by studying

the slow motion nonlinear

dynamics of the system.

Stability analysis and bifurcation

theory are used to determine the changes in SMS dynamics

in deep water based as

functions of buoy parameters.

Catastrophe sets in a two-dimensional parametric design

space are developed from bifurcation boundaries, which

separate

regions

of qualitatively

different

dynamics.

Stability analysis defines the morphogeneses occurring

as bifurcation boundaries are crossed. The mathematical model

of the moored vessel consists of the horizontal plane

-surge. sway and yaw -

fifth-order, large drift, low speed

maneuvering equations.

_{Mooring lines made of chains are}

modeled quasistatically as catenaries supported by buoys

including nonlinear drag and touchdown.

Steady excitation

from current, wind and mean wave drift

are tnodeled.

Numerical applications are limited to steady current and shOw

that buoys affect

_{both the static and dynamic loss of}

stability

of the

sYstem, and may even cause chaotic

response.

NOMENCLATURE

AC anchored catenary

CG center of gravity

D8 buoy diameter

FPSO Roatin Production Storage and Offload

h water depth

depth of AC

depth of segment S2A of SC depth of segment S2B of SC L

length of vessel

r

IECNNI$CHE UP1fl1ERSngg

Laboratorium voor

SCheepshydrmhI

rchief

Mekelweg

_{2, 2628 CD Delit}

'ICL016- 1868/s . Fa 015 78183S

EFFECT OF SIZE AND POSITION OF SUPPORTiNG BÔYS ON

THE DYNAMICS OF

SPREAD MOORING SYSTEMS

Luis 0. Garza-Rios

Michael M. Bernitsas

Department of Naval Architecture

and Marine Engineering

University of Michigan, Ann Arbor,Michigan

total length of the mooring line (1wt = +

total lencth of AC

total length of SC/location of buoy along the mooring line

mass of vessel mass of buoy

number of mooring lines number of buoys

anchored catenary segment suspended catenary segment

segment of suspended catenary running from the buoy to the lowest point of SC

segment of suspended catenary running from the vessel to the lowest point of SC suspended catenary

Spread Moonng S)stem

horizontal tension components of SI (AC) horizontal tension components of S2 (SC) current speed

inertial reference frame/position of vessel CG with respect to inertial frame

position of the center of gravity of buoy with respect to the inertial frame

vessel fixed coordinate system buoy fixed coordinate system

coordinates of the anchoring point oat the sea floor

body fixed coordinates of the vessel faii'léad current angle with respect to (x,v,:) frame horizontal angle between the X-axis and the mooring line. In m8 n SI S2 S2A S2B SC SMS Uc (x.v,z)

(x5,8.z8)

(X.Y.Z) (X3. Y8. ZB)

a

Q

I.

INTRODUCTION

Station-keeping of ships and other floating production systems cart.' be achieved via mooring and dynamic positioning. In recent years. the

design problem of

moorirg FPSOs has gained considerable attention as the water depth of oil reserves, gas and other natural 'resources has increased. Presently. station-keeping in deep waters

(700 -

1800 meters) via FPSOs is generally achieved by dynamic positioning 15, 19]. while mooring takes place mostly in shallow and intermediate water depths (less than 700 meters). The next logical step toward achieving an efficient and less expensive station-keeping design would be to develop a hybrid system with dynamic positioning and mooring. To design sound mooring systems in deep waters, however, it is necessary to gain a qualitative understanding of their slow motion dynamics in such an environment.

Several studies show that mooring systems dynamics strongly depend on certain design parameters, such as the number of mooring lines; the fairlead position, material, orientation, pretension of lines, etc. [1-4. 6. 7,

10, II. 13.

16, 18, 20]. A preliminary study of the nonlinear dynamics of Turret Mooring Systems (TMS) in relatively deep waters (on the order of 1200 meters) without buoy supports [7] shows that the system tends to lose its dynamic stability with increasing water depth. In deep waters, the weight

exerted by the mooring lines on the

vessel and onto themselves may lead to high stresses on the vessel and mooring line breakage. This problem can be solved by

supporting the line weight by buoys, utilizing different

types of lines (such as chains, synthetic fiber ropes, and steel cables), or a combination of both. In this work, the dynamics of mooring systems with anchoring chains and supporting buoys are studied, and the effects of the position and size of the buoys on the system are assessed based on the design methodology for mooring systems developed at

the University of Michigan since 1983 [2, 3, 4,

10, 18].

Section 11 describes the mathematical model of the system, mooring line model, and external excitation. The state space representation of a spread rhooring system with supporting buoys, and the equations to solve for system equilibria are derived in Section III. In Section IV. stability analysis is performed and bifurcation theory used to assess the behavior of two different geometries of a line, four-buoy system. Catastrophe sets around the principal equilibrium position are constructed to discern the system

dynamics for different water depths and buoy size and

position along the mooring lines. These sets reveal that the effects of buoy size and position on the system dynamics vary widely with system geometry.

In Section V. the

feasibility of mooring in deeper waters with buoy supported chains is discussed.

Ii. PROBLEM FORMULATION

The slow-motion dynamics of a Spread Mooring System (SMS) with chain mooring lines and their supporting buoys

is modeled in this section in terms of the equations of

motion and the kinematics of the vessel and each supporting buoy, the mooring line catenary equations of the system,

and external excitation consisting of time independent

current, wind and mean wave drift. The vessel hvdrodynamic maneuvering model is based on the large drift angle. low speed. fifth-order maneuvering equations [21. 22]. The

equations of motion of the system consist of the horizontal plane - surge. sway and yaw - slow motions for the vesel. and motions or each supporting buoy in surge. sway nd heave. First the mooring line model with one supporting

buoy is derived.

11.1.

Mooring Line Model

The mooring lines of the system are model'ed quasistatically as a submerged catenary partially supported with buoys. To facilitate discussion, the concepts describ'ed in this paper apply to SMS with a single buoy supporting each cacenary. A complete SMS mathematical model with any number of buoys per line is derived by Garza-Rios and Bernitsas, 1999 [81.

Figure 1 shows the general geometry of a submerged twp. dimensional catenary whose weight is partially supported b a single buoy in water depth h. The inertial frame (x.z has its origin at the anchoring point on the ocean floor. The catenary is divided into two segments. Segment SI consists

of the

portion

that spans from the

mooriIg terminal on the ocean floor to the supporting buoy, and is

labeled as AC (Anchored Catenary) [9].

Segment S2 connects the buoy to the floating body. is labeled as SC (Suspended Catenary) [17]. and is further divided into segments S2A and S2R. as shown in Figure 1. Segment S2A runs from the point of connection of the buoy to the Iowest point of the SC. and segment S2B runs from the lowest point 'of the SC to the point of attachment on the vessl. The geometric properties of the catenary shown in Figure 1

are: P. the total length of the horizontal projection of AC:

. the length of the horizontal projection of the suspendd

part of AC: d. the horizontal distance of AC in the ground:

1,. the length of the horizontal projection of SC: '., tte

length of the horizontal projection of segment S2A:

f,5,

the length of the horizontal projection of segment S2B; h. the vertical distance of the lower point of the buoy from th'e sea bed.. is also the depth of AC; h4. the vertical distanc from the lower point of SC to the lower point of the buoy. is also the depth of S2A: and ''B' the vertical distance from

the lower point of SC to the point of attachment to th

floating vessel, is also the depth of S2B. The following relations hold:

': =F4+P5

h=hIh,A+h.B

The total length of the catenary ('wT) is the components F arid F.. which correspond to of segments SI and S2. respectively..

= 1w1 +

The external forces acting on the catenary of Figure I are

[8]: F05 is the buoy drag force due to current: drag force on segment SI of the catenary: F.

F!D

and

is the,

F:8D

are

the drag components on segments S21 and S2B,

respectively; and F5. is .the force imposed by the floating:

(1 (2) (31

sum of its

the lengths. (4)

vessel on the

_{moorinc line at the}

point of attachment .tôthe

vessel (an end-point condition).

_{In addition. L!.is the}

current speed

acting in the - .r

direction.

The tension in each of the catenary

end points is given by

r.=.j72.+12.

(5)

where T. is the tension of segment S. and 1, and 7.

are

the horizontal and vertical tension components of segthent

S.. respectively. The

horizontal tension components of

segments SI and S2 are therefore denoted by 7

_{arid 7,.}

The horizontal tension components of segments S2A and

S28 are equal in magnitude. The vertical tension component

of each of the three segments is.given by [8]:

112.

_{Fquations of. Motion}

The catenary model presented in Subsection 11.1 is used

here to model the line forces exerted on the floating vessel.

Figure 2 shows the general geometry of a n-line SMS

showing 2 + B reference frames, where n8 is the number of buoys in the system and 11

_{is the number of mooring lines}

(B =

n).

In Figure 2, (.t.v. ) is the inertial reference frame

with its origin located at mooring terminal

1; (X,Y,Z) is the vessel-fixed reference frame with its origin located at the

center of gravity of the vessel (CG); (X.Y'.Z) is the

buoy-fixed reference frame of the i-ih (1

= 1,.., n5) buoy.

with X

pointing horizontally

in the

direction of its

anchoring point (x,v.:), and Z

pointing upwards.

Figure 2 also shows that the deformation of the two

seements of each line, AC and SC. is two dimensional. Due

to the drag forces acting on the buoys, however, the two

segments of each line are in different vertical planes making the overall deformation of each line three dimensional.

The slow motion dynamics of the system can be modeled

by three equations for the horizontal plane slow motions of

the vessel in surge. sway and yaw, and three equations of

motion in surge, sway and heave for each buoy in the

system.

The horizontal plane equations of motion of the vessel in

surge. sway and yaw are given by

(m+m)u(m+ni)rt

=XH +

{lcos/3w

-+ 1C'surge ₍₉₎ = +

j{7sini3'i

-

F}

(6) =

7,.

sinh[.2LJ.

₍₇₎ =7

Siflh[.B)

₍₈₎ + Fcwa) (lOp

(L

+ J. )r

=_{NH +}

_{X{iY sin/3m}

_-

F.}

-

.

(II)

1=1

where in

is the mass of the vessel; ,n and m are the added

mass terms in surge and sway; L is the second moment of

inertia about the Z-axis;

.1... is the added moment of inertia: u. v,

and ii, v are the relative velocities and accelerations

in surge and sway with respect to water;

r

and r are the

relative angular velocity and acceleration in yaw with respect to water.

For each mooring line in equations (7) - (9). (xe.

'ce) are the vessel-fixed coordinates of the mooring line

fairlead;

_$

is the. horizontal angle between the X -axis and the mooring line measured counterclockwise from the vessel: 'BDx and F.80 are the .drag forces on segment 2B of SC in

the X and Y directions, respectively [8, 9, IS].

In equations (9)-ill).. X.

H' N

are the

velocity

dependent horizontal plane hull hydrodynaniic forces and

moment in surge. sway and yaw expressed in terms of the

large drift, slow motion derivatives. [10, 21. 22) as:

XH = X,, +Xuu+LXuuu2

_+X.vr

₍₁₂₎

= }.v + +

}.v5

+ Yurur+_Y,.urIr1+_1cjrlITI (13) Nfl

.'sv + Nuv +

+ +Nrr

+ Ndd

+N1,juvIr+ N.v2r

(14)

In addition. Fsurge. and are the external forces

and moment acting on the vessel due to steady wind and

second order mean wave drift

[3].

These forces are not

considered in the numerical applications in this study since

the focus here is the effect of buoys on SMS design.

The vessel kinematics are given by

.i= ucosi' -

+1]. cosa ,

(15)

V=usinW+vcosW+USina ,

(16)

(17)

where (J is the current speed; a is the current angle with

direction as shOwn in Figure 2. and

'

is the drift angle of

the vessel.

The equations of motion for each of the supporting buoys. in surge. sway and heave are given by [8]:

(m

+.4i" =

7j +Tjcos$'

F'F.,

. (18)

(m'

+

)V = Tj'

cos$

-

-

-

F,1

. (19)

p,(i) (m + .4

)(I)

= F" - 7;

sinh(__y5_J

(piJ '

(I)

" f)

for 1=1.

For

each buoy in the system, m9 is the buoy mass; A11,

A,, and A33 are the buoy added mass components in surge,

sway and heave; u8, v8 and w8 are the buoy relative accelerations in

surge. sway and

heave: F8 is the net

buoyancy

force of

the buoy, i.e. the difference between its

buoyancy and its weight; F0, and FDBY are the drag forces

on thebuoy in surge and sway [8J; F4, and LI are the drag forces on AC in the directions parallel and perpendicular to

the mooring line motion [8]; F,A and F.ADy are the drag

forces on segment S2A Of SC in the X3 and Y9 directions, respectively [8]; and $8 is the horizontal angle between the two segments of the catenary, measured coUnterclockwise

from X8.

The 3 n8 associated

kinematic relations of the buoy

system are [8J:

i=ucosyvsiny+U!_cosa

(21) (I) (iI

' =usiny+vcosy+U!L_sina

(22)

.(i)_

(I) (23) '-8

for i=l...n8.

In equations

(21) - (23),

UB. VB and w8 are the relative velocities

of

the buoy in surge, sway and heave; YB is the

horizontal angle between the x-axis and Sl, measured

counterclockwise from the buoy: and z8 is the vertical ordinate of the center of gravity of the buoy. Notice that kinematics (21) - (23) assume a linear current profile.

III. STATE SPACE REPRESENTATION AND

EQUILIBRIA OF SMS

The nonlinear mathematical model for the buoy supported SMS in deep water presented in the previous section consists, of the following 6(1 + n8) equations: three equations of motion for the vessel in surge, sway and yaw (9)-(1 1); 3 n8 buoy equations of motion in surge. sway and heave (18)-(20): and their respective 3+3 n8 kinematic relations (15)-(17) and (21)-(23).

111.1.

State Space Representation

The equations of motion and kinematic relations of the

system can be recast as a set of first order nonlinear coupled differential equations of the form

by selecting the following state space variables:

(20)

_{x =[u.}

_{. r ..v. r. ty,}

v,

_{x'. r'. .LJI}

(m

+A)

-(I)

.=ucosy_vsiiyUL_cosa

=usiny +vcosy +Ucsina

_(iJ (i)

_8

for i=l...

All . variables in equations (26) - (37) are direct or indirect functions of the state space. vector field '25).

111.2.

Equilibria of SMS

The equilibria of the nonlinear model of SMS are

stationary flows of vector

field . X. and correspond to

singular points of equation (24). Equilibria can be found by setting the time derivative of the state variable vector to

zero, i.e.

(27

(28.)

i=l ...n8,.

(2)

Then, the nonlinear model for the SMS with supporting buoys takes the form:

XH + (rn + ,n,,)rv

{i

cos$'

_{F,. } + F;ure} 1 =

(m+m)

(26 - (m + m )ru +

{i

sinp(s)

- F,,, } +

---

(m+m)

n NH +

x{isin$')

_-

_12BD1} - i=!

--(l. +J)

cos$

_{cb }}

(I__ +J..) x = ucosyi - sini +U cosa

=usinW+vcosy/+Usina

yF=r.

7 + 7 cos$' F

-(m

+A)

T4i) - (t j) (1) (i) COS -R - DR) - LI .4DY B

-

(rn+A)

F' -, TJ

sintI__]

Pe°

7sinh[_L'

pç(i)

(j

i=f(x).

(24) e is an equilibrium of

where the overbar on the state variable vector i denotes an

equilibrium state.

The number of equations that must be solved

simultaneously to find an equilibrium constitute the zero solution

to equations (26).37).

The so!ution to such equations, however, requires that the tension components 1 and

i,,

be known. Therefore. additional auxiliary equations are needed to solve for the tension components required in the equations of motion and the geometrical properties of the catenaries [8. Due to state space equations (29) and (30). the equilibrium values for the vessel velocity components u and

v can be

recast

in terms of the

equilibrium drift angle t,v as f011ows:

u=ULcos(i7a)

. (39)

7=Usin(i7a)

. (40)

Furthermore, state space equations (31) and (37) reveal that the values for the vessel rotational velocity r and the buoy

heave velocities w, (i=l,

equilibrium; i.e.

NH

+x

{jsinW

_TU}

ye,

_v{T

cos/3"

_-

₊ caw = 0

where it is implied that the functions containing i can be replaced by relatioris (39) - (41). The buoy equilibrium equations become:

(i)

_I

7(1 ')

-

(1r)1

[.L.J

iflh_[

:i)

n8), are zero

at (45)

for i=1 ...n8.

Thus. 3 +1 1 ri _{equations must be solved for equilibrium. In}

expressions (51) and (52). it has been assumed that the

vertical position of the buoy center of gravity

z is

approximately equal to the depth of its corresponding AC (h'). The relation between these is given by

R + ₍₅₇₎

where is the radius of the i-th buoy.

In equations (55) and (56). for mooring line i. (.x. v)

are the horizontal plane coordinates of the anchoring point

on the sea floor: and (i!,

.t).) _{are the horizontal plane} attachment coordinates on the vessel with respect tO the inertial reference frame (x. v).

d

IV. STABILITY AND BIFURCATIONS OF

EQUILIBRIA

Equilibria

_{of nonlinear dynamical}

_{systems and the} behavior of such systems near equilibrium provide local information on dynamical flows (trajectories) [12]. Such information can be useful in surmising the global beiàvior

of a system

in _{the time domain or in a phase space.} Consequently. knowing all equilibria and - their nature provides global information ol the qualitative behavior of the system and renders time simulations _{unnecessary in} general [1, 3. 4. 7. 18]. Further, bifurcations of equilibria

for i=I...

The auxiliary equations (43)

sinh(..!)]

1 7(1) '\ sinhi Icoshi

-(

n9. equations (50) are

I

7(i)'\

_____,JO(zW+2__)_o

that must [8): 1 7(i

I+coshl

.4-t.27

₎

be solved

(

(j) Isinhi

-4k-(27j

along with (-51) 0 (52) (53) (54) (55) (56)

27)

(i) e° _.._.i_sinhI4 P (i)

----sinhI

U-

p

7(1) \

27)

I \

__'

\

=o.

(I)

I

7(i) 2B

sinhI

I

7(t) 2A i;:

)

P

(27)

((ii)2 +(1)i))2

-7

ii't sin ?' +

t)cos) +U-

sna = 0

(50

Thus, the 6(1 + n8) state space equations that must be solved

simultaneously to find the SMS equilibria can be reduced to 3+5 n8- cquations. In

addition.

_{6B auxiliary}

tension/geometry equations need tà be solved in conjunction

with the

remaining state space equations to

find an

equilibrium [8].

The equilibrium equations for the-vessel thus become:

+ _{?O cos

-

+ urge =0 (43)

and their corresponding morphogeneses make it possible to

identify areas of qualitatively acceptable dynamics.

Th.0

eliminates the need for trial and error in design of mooring

systems.

Based on

this

philosophy and

numerical

applications we assess in this section the effects of buoy

size and position on the design of SMS in deep water.

IV.1.

Theoretical

Considerations

Nonlinear stability

analysis is

used to

discern the

dynamics of specific SMS/buoy configurations.

The

stability

properties of the system can be obtained by

performing eigenvalue analysis around all system equilibria.

If all eigenvalues of the system have negative real parts, a

particular equilibrium is stable, and all trajectories initiated

near that equilibrium will converge toward it asymptotically.

If at

least one eigenvalue has a positive real

part. that

equilibrium will be unstable and a small disturbance from

equilibrium will cause the system trajectories to diverge from it t23).

Bifurcation sequences are then studied to find

qualitative changes in the dynamic behavior of the system

by varying one or more design variables for a specific

mooring system configuration.

Such sequences determine

regions of qualitatively similar behavior, such as stable.

unstable. periodic, and chaotic. Graphically they are plotted

in stability charts. commonly referred to as catastrophe sets.

In this paper, catastrophe sets are constructed as functions

of two parameters:

the diameter of the buoys (D3). and the

location of the buoy along the mooring line, defined by

t,,2.

_{All buoys have the same diameter and all are placed at}

the same location along the corresponding line.

The water depth h

is used as parameter in catastrophe sets

for

h =700.

750 and 800 m.

The ranges of buoy size and its

position along the mooring line depend on a number of

factors such as the water depth and the net buoyancy force of

each buoy.

Thus, for a particular chain mooring line in a

specific water depth. only a certain range of combination of

buoy size and position can be used effectively to support the

weight of the chain.

In this paper, rather than providing a

complete study of size-location combination for buoys. we

determine bifurcation sequences for specific ranges of those

parameters.

This allows us to assess the effects of the

supporting buoys on the dynamics of SMS.

IV..2.

Analysis of the SMS

To illustrate the possible effects of buoy size and position

on the dynamics of the system, we consider a 4-line tanker

SMS whose geometric properties are shown in Table

1 [10)

in a 2 knot head current, a

180'.

The dimension of the

state space is 6(1 + n5).

For a 4-line SMS with one

supporting buoy per line.

n8 = n =4, and the state space

becomes 30-dimensional.

In

this example, all moonng

lines have the same length

',. = 1500 meters. with an

average breaking strength of 5159 kN [14).

In addition. to

different geometries of the SMS (denoted as UI and G2 are

considered, and their characteristics shown in Table 2.

The

tanker is moored in a symmetric pattern with its (X.Y)

a.'us

parallel to the (x.v) reference frame as shown in Figure 3.

The orientation of each line in Table 2 is defined by £7. the

horizontal angle between the vessel-fixed X -axis and the

mooring line measured counterclockwise.

Table I. Properties of the tanker SMS Property

LOA (length overall) LWL (length of waterline)

B (beam)

D (draft)

C8 (block coefficient) 1 (displacement) m m

I--

J.-272.8 m 259.4 m 43.1 rn 16.15 m

0.83

I.5374x105 tons

9.110x106 kgs

1.360x

108 kgs

7.180x 1011 kg.m2

5.430xl0H kg.m2

Table 2 and Figure 3 show that the only difference

between geometries GI and G2 is in the aperture between th two forward mooring lines, which is bigger in geometry G2. Also note that the horizontal pretension component for both

segments of each catenary are equal.

These are based ot

pretension values for the pair D8 =7 m.

f,

=700 m.

Figure 4 shows a series of catastrophe sets

for th& principal

equilibrium position of the system for

both

geometries GI and G2 in the parametric design space (D5

w2) with the following ranges

5mD8 7m.

600 m S F,,, 5 750 m .

I

fOr water depths of 700. 750 and 800 meters.

Table 2. Characteristics of SMS geometries GI and G2 Characteristic Geometry GI Geometry G2 Fàirlead Coordinates (m)

x. v

42), 2)

The catastrophe sets of Figure 4 present three regions of

qualitatively different dynamics, and are labeled as I, hand

Ill.

The three lines on the right correspond to bifurcation

£7(l) _350.00

_315.00

10.00

45.00

(3) _{170.00 170.00} £714) 190.00 190.00 Horizontal Pretension (kN) Line I 260.5628 260.5628 Line 2 260.5628 260.5628 Line 3 104.2251 104. 225 1 Line 4 104.225 I 104.2251

125.00. -4.1023

125.0. -4.1023

125.00, -4.1023

125.0. -4.1023

-129.69. 0.00

- 129.69, 0.00

-129.69. 0.00

-129.69, 0.00

Orientation of lines (deg)

boundaries for geometry GI: the remaininc two lines on the

left correspond to bifurcation boundaries of geometry G2.

Thus, regions Gl-1 and GI-Il pertain exclusively to geometry

GI. while regions G2-1 and G2-lIl pertain to geometry G2 in

Figure 4. These regions are described as follows:

Region I: The principal equilibrium of the system is stable; all 30 eigenvalues have negative real parts. A small random

disturbance

from

the

principal

equilibrium

initiates

trajectories converging to it in forward time.

Region II:

The principal equilibrium of the system is

unstable with a one-dimensional unstable manifold. i.e.

it

has a real eigenvalue with positive real part.

The system

undergoes static loss of stability when crossing from region

I to region II. with the genesis of two additional stable

equilibria.

These may be stable

or unstable for the

parametric ranges shown in Figure 4 depending on the size

and position of the buoy.

Stability and bifurcation analyses

around these equilibria is, however, out of the scope of this

paper.

Region lii:

The principal equilibrium of the system is

unstable with a four-dimensional unstable manifold.

Two

complex pairs of eigenvalues have positive real parts.

This

corresponds to dynamic loss of stability of the two forward

buoys when crossing from region I to region Ill.

Since there are no other nearby equilibria to attract the trajectories. the forward buoys oscillate. Since the motions of the buoys

and the vessel are coupled. the system itself exhibits chaotic

motions due to the

large amplitude oscillations of the

forward buoys.

In simulations this

is manifested in the

chaotic motion of the vessel.

In region Ill, the dimension

of the unstable manifold is four.

The dynamics exhibited by each geometry in Figure 4 is

detailed below:

Geometry GI:

As shown in Figure 3, geometry GI is

characterized by the small angle of aperture between the two

forward mooring lines (20').

The sets of three lines at the

right of Figure 4 correspond to static bifurcation boundaries

of geometry GI for water depths of 700, 750 and 800

meters.

As previously mentioned, the pnnëipal equilibrium

of the system loses its static stability when crossing from

region I (stable) to region II (unstable), and two alternate

stable equilibria result from such bifurcation.

Note that for

the ranges of parameters shown, stable region

I increases

with increasing buoy position

_{( '. buoy size (DB) and}

water depth (h).

Geometry G2

This geometry is characterized by a, large

angle of aperture of the forward mooring lines (90') as

shown in Figure 3. and thus

it is

expected to exhibit

qualitatively different dynamics with respect to geometry GI.

The sets of two lines to the left of Figure 4 correspond to

dynamic bifurcations of geometry G2 for water depths of 750

and 800 meters.

Specifically, these lines denote boundaries

of dynamic loss of stability of the forward buoys in the

system, and occur when crossing from region I to region Ill.

The principal equilibrium of the system is thus stable to the

right of its corresponding bifurcation line and unstable

(chaotic) to the left.

Since there are no other equilibria to

attract

the

trajectories,

the

system undergoes chaotic

(unpredictable) motions resulting ultimately

in line breaking.

Notice from Figure 4 that such unstable region

increases as the position and size of the buoy decrease, and

as the water depth increases.

_{As shown in Figure 4, the}

system

is'stable for all

( DB. Ps,,) values for a water depth of 700 m.

A preliminary conclusion of the dynamics of the

buoy-supported SMS is that geometry GI renders a more stable

configuration, since the unstable region leads to alternate

stable equilibria, while geometry G2 may result in chaotic

behavior. The selected ranges of parameters (D5. t,.,. h).

however, lead to wide variations in the depths of each

mooring line segment (h1 for segment SI, h2A for segment

S2A, and h8 for segment S2B), and therefore variations in

the mooring line tension. Thus, it may be impractical to try

to design a mooring system based on all parameter ranges

for the pair (D9, ',) in Figure 4.

Table 3 shows the

principal equilibrium values of the various depths h1 . h.,4

and h

for both geometries Gl and G2 at the four end

points of Figure 4, for a water depth of 750 m. Due to the

symmetric nature of the system, the equilibrium values of

these variables are equal for Mooring Lines (ML) I and 2:

the same holds true for Mooring Lines (ML) 3 and 4.

Table 3. Equilibrium values for vertical distances of each catenarv segment (rn). h =750 m

One design goal. for a buoy is that the depth of neither

catenarv segment becomes equal to the water depth. U that

were to be the case. there would be no need to implement

buoys in the system. As shown in Table 3. neither calenary

segments depth reaches the water depth

at

any one

Geometry GI Geometry G2

\ILI and 2 ML 3 and 4

ML I and 2 ML 3 and 4

(DB, ',) (m)

182.279

= (5.0,600.0)

184.771 188.033 184.754 h.1

0.179

1.149 3.120

0.940

h,B 567.901

566.378

565.070 566.186 (DB.

w2) (m) = (5.0,750.0)

h1 .121.391 124.119 _{118.622 125.247} h 19.530 27.101 9.679 30.007 648.2 19

652.982

641.057 654.760

(P8. ') (m) = (7.0,600.0;

496.815

504.623

485.993 511.548 h

94.207

102.619 72.440 115.619 h

347.392

347.997

336.447 354.070 (D5, h1 2) (m)

436.595

= (7.0.750.0)

443183

424.823 451.128 '2A

120.096

128.419 95.357 144.126 h,3 _433:5b1

435.237

420.534 442.998

parametric end point in Figure 4. Notice from Table 3 that with increasing distance the weight that each buoy must support increases as well. For a particular buoy size. this leads to a decrease in the '.ertical distance of the AC

(h1) and an increase in the depths of S2A and S2B.

Conversely, as the size of the- buoy D9 increases for a particular value of P, the net buoyancy of the buoy also

increases, thus providing an added upward force and

increasing the depth of AC. This also increases the depth of S2A (hzA) and decreases the depth of S2B( h,5). Table 3 also shows that at the two end points of Figure 4 for DB=5

m, the depths of segments S2B are excessively high. The same holds true for end point (DB

i7 m.

e,,, =600 m) for

the values of h1. - The end point (D8=7 m. _',,2=750 m) results in well distributed values for the various depths at equilibrium, and thus a design based on these values would be more practical. Also note from Table 3 that the lower

left corner end point

( D =5 m,

es.. =600 m) results in small values for the depth h,A. In simulatiOns, the large

motions of the vessel will cause the system to lose the

geometry of Figure I. In such case, the nature of the suspended catenary is lost, resulting in excessive mooring line tensions.

Other combinations of buoy size/position in Figure 4 may result

in inadequate mooring line

tensions, which are

proportional to the depth of the catenary.

The most practical values for DR and _, therefOre lie within some specific ranges of the parameters considered in Figure 4. Table 4 shows the equilibrium values of h1 . h and

h5

for

both geometries G I

and 02 for the pair

(D8=6.85 rn.

1w2 =695 m) at a water depth of 750 m. Notice thaL the equilibrium values for the depths of the segments are less than those in Table 3 for the pair(DB=7 m, _'w2 =750 m).

This results in lower vertical mooring line tensions and

therefore are more practical. In addition, this parametric configuration is stable for both geometries, as shown in Figure 4 Other combinations of the parameters

_(D8,

₂₎ may result in more practical geometric configurations as well. The stability and bifurcation analysis presented in this section. however, should be used to determine whether such geometries result in sound stable systems.

Table 4. Equilibrium values for vertical distances (m). D5=6.85 m. 695 m, h 750 m

Geometry 01 Geometry 02

The results obtained in Tables 3 and 4 indicate that the practical values for the pair (D8, f,) exclude, for the most part, all ranges in the lower and left sides of Figure 4. For the remaining ranges, geometry 02 shows stable behavior of

its principal equilibrium, while geometry G I may fall in

static loss of stability, resulting in a non-svmmetric equilibrium. The design of a buoy supported SMS based n either -geometry (or any other system conliguration and catastrophe sets would require further analysis based on simulations to assure that the maximum limits of the vessel motions and the mooring line tensions are satisfied.

V. CONCLUDING REMARKS

-A mathematical model has been developed for modeli9g

the slow motion horizontal plane dynamics of moorirg

systems with supporting buoys: The design methodology developed at the University of Michigan fOr spread mooridg systems [1-4 10. 18] has been extended to include buoy.

This methodology makes it possible to assess the effects of buoys on the system behavior without trial and error dr repealed simulations. It is

based on deriving

tim1e

independent properties of mooring systems with buoys such as equilibria and their stabilities, bifurcation sequences an$i their singularities, and morphogeneses across bifurcation boundaries. Numerical applications of a 4-line FPSO with one buoy per mooring line have been used to illustrate this methodology and assess the importance of buoys in mooring dynamics. It has been shown that variations in position and size of supporting buoys may lead to static and dynamic loss of stability and even to chaos, depending on the geometry of the. system. The effect of water depth on the systerii dynamics has also been studied, and it was shown that the dynamically unstable region increases with increasing wate depth. while the opposite effect is observed for staticalh unstable systems.

ACKNOWLEDGMENTS

This work is sponsored by the

University

o:

Michigan/Industry Consortium in Offshore Engineering. anc Department of Naval Architecture and Marine Engineering

University of Michigan.

Industry participants includt Amoco. Inc.; Conoco. Inc.: Chevron. Inc.: Exxor Production Research; Mobil Research and Development Shell Companies Foundation; and Petrobrás, Brazil.

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Figure 1. Geometry of buoy-supported catenary

GEOMETRY Gi

Figure 3. Geometric characteristics of Ci and G2

hgure -4. _{Catastrophe set of buoy supported SMS - Geometries Gi and G2}

750 725-700-'. 675 -650

625 -\

---G2

S S. S. --

HI \ G2-- i

S. -S.

\\

\

-\.

\

\

'

\

S.

G I - 11

S. S. S. S.

G i

-.5

\\

I.

i

..

_:_

h - 700 m h-7-50m h-800-rñ

---DB (m) 0 52 54 56 60 612 64 616 68